Internet Draft D. M'Raihi
Category: Informational VeriSign
Document: draft-mraihi-oath-hmac-otp-04.txt M. Bellare
Expires: April 2005 UCSD
F. Hoornaert
Vasco
D. Naccache
Gemplus
O. Ranen
Aladdin
October 2004
HOTP: An HMAC-based One Time Password Algorithm
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Abstract
This document describes an algorithm to generate one-time password
values, based on HMAC [BCK1]. A security analysis of the algorithm
is presented, and important parameters related to the secure
deployment of the algorithm are discussed. The proposed algorithm
can be used across a wide range of network applications ranging
from remote VPN access, Wi-Fi network logon to transaction-oriented
Web applications.
This work is a joint effort by the OATH (Open AuTHentication)
membership to specify an algorithm that can be freely distributed
to the technical community. The authors believe that a common and
shared algorithm will facilitate adoption of two-factor
authentication on the Internet by enabling interoperability across
commercial and open-source implementations.

1. Overview
The document introduces first the context around the HOTP
algorithm. In section 4, the algorithm requirements are listed and
in section 5, the HOTP algorithm is described. Sections 6 and 7
focus on the algorithm security. Section 8 proposes some extensions
and improvements, and Section 9 concludes this document. The
interested reader will find in the Appendix a detailed, full-fledge
analysis of the algorithm security: an idealized version of the
algorithm is evaluated, and then the HOTP algorithm security is
analyzed.
2. Introduction
Today, deployment of two-factor authentication remains extremely
limited in scope and scale. Despite increasingly higher levels of
threats and attacks, most Internet applications still rely on weak
authentication schemes for policing user access. The lack of
interoperability among hardware and software technology vendors has
been a limiting factor in the adoption of two-factor authentication
technology. In particular, the absence of open specifications has
led to solutions where hardware and software components are tightly
coupled through proprietary technology, resulting in high cost
solutions, poor adoption and limited innovation.
In the last two years, the rapid rise of network threats has
exposed the inadequacies of static passwords as the primary mean of
authentication on the Internet. At the same time, the current
approach that requires an end-user to carry an expensive,
single-function device that is only used to authenticate to the
network is clearly not the right answer. For two factor
authentication to propagate on the Internet, it will have to be
embedded in more flexible devices that can work across a wide range
of applications.
The ability to embed this base technology while ensuring broad
interoperability require that it be made freely available to the
broad technical community of hardware and software developers. Only
an open system approach will ensure that basic two-factor
authentication primitives can be built into the next-generation of
consumer devices such USB mass storage devices, IP phones, and
personal digital assistants).
One Time Password is certainly one of the simplest and most popular
forms of two-factor authentication for securing network access. For
example, in large enterprises, Virtual Private Network access often
requires the use of One Time Password tokens for remote user
authentication. One Time Passwords are often preferred to stronger
forms of authentication such as PKI or biometrics because an
air-gap device does not require the installation of any client
desktop software on the user machine, therefore allowing them to
roam across multiple machines including home computers, kiosks and
personal digital assistants.

This draft proposes a simple One Time Password algorithm that can
be implemented by any hardware manufacturer or software developer
to create interoperable authentication devices and software agents.
The algorithm is event-based so that it can be embedded in high
volume devices such as Java smart cards, USB dongles and GSM SIM
cards. The presented algorithm is made freely available to the
developer community under the terms and conditions of the IETF
Intellectual Property Rights [RFC3668].
The authors of this document are members of the Open AuTHentication
initiative [OATH]. The initiative was created in 2004 to facilitate
collaboration among strong authentication technology providers.
3. Requirements Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in
this document are to be interpreted as described in RFC 2119.
4. Algorithm Requirements
This section presents the main requirements that drove this
algorithm design. A lot of emphasis was placed on end-consumer
usability as well as the ability for the algorithm to be
implemented by low cost hardware that may provide minimal user
interface capabilities. In particular, the ability to embed the
algorithm into high volume SIM and Java cards was a fundamental
pre-requisite.
R1 - The algorithm MUST be sequence or counter-based: One of the
goals is to have the HOTP algorithm embedded in high volume devices
such as Java smart cards, USB dongles and GSM SIM cards.
R2 - The algorithm SHOULD be economical to implement in hardware by
minimizing requirements on battery, number of buttons,
computational horsepower, and size of LCD display.
R3 - The algorithm MUST work with tokens that do not supports any
numeric input, but MAY also be used with more sophisticated devices
such as secure PIN-pads.
R4 - The value displayed on the token MUST be easily read and
entered by the user: This requires the HOTP value to be of
reasonable length. The HOTP value must be at least a 6-digit value.
It is also desirable that the HOTP value be 'numeric only' so that
it can be easily entered on restricted devices such as phones.
R5 - There MUST be user-friendly mechanisms available to
resynchronize the counter. The sections 6.4 and 8.4 detail the
resynchronization mechanism proposed in this draft.

R6 - The algorithm MUST use a strong shared secret. The length of
the shared secret MUST be at least 128 bits. This draft RECOMMENDs
a shared secret length of 160 bits.
5. HOTP Algorithm
In this section, we introduce the notation and describe the HOTP
algorithm basic blocks - the base function to compute an HMAC-SHA-1
value and the truncation method to extract an HOTP value.
5.1 Notation and Symbols
A string always means a binary string, meaning a sequence of zeros
and ones.
If s is a string then |s| denotes its length.
If n is a number then |n| denotes its absolute value.
If s is a string then s[i] denotes its i-th bit. We start numbering
the bits at 0, so s = s[0]s[1]..s[n-1] where n = |s| is the length
of s.
Let StToNum (String to Number) denote the function which as input a
string s returns the number whose binary representation is s.
(For example StToNum(110) = 6).
Here is a list of symbols used in this document.
Symbol Represents
-------------------------------------------------------------------
C 8-byte counter value, the moving factor. This counter
MUST be synchronized between the HOTP generator (client)
and the HOTP validator (server);
K shared secret between client and server; each HOTP
generator has a different and unique secret K;
T throttling parameter: the server will refuse connections
from a user after T unsuccessful authentication attempts;
s resynchronization parameter: the server will attempt to
verify a received authenticator across s consecutive
counter values;
Digit number of digits in an HOTP value; system parameter.
5.2 Description
The HOTP algorithm is based on an increasing counter value and a
static symmetric key known only to the token and the validation
service. In order to create the HOTP value, we will use the

HMAC-SHA-1 algorithm, as defined in RFC 2104 [BCK2].
As the output of the HMAC-SHA1 calculation is 160 bits, we must
truncate this value to something that can be easily entered by a
user.
HOTP(K,C) = Truncate(HMAC-SHA-1(K,C))
Where:
- Truncate represents the function that converts an HMAC-SHA-1
value into an HOTP value as defined in Section 5.3.
The Key (K), the Counter (C) and Data values are hashed high-order
byte first.
The HOTP values generated by the HOTP generator are treated as big
endian.
5.3 Generating an HOTP value
We can describe the operations in 3 distinct steps:
Step 1: Generate an HMAC-SHA-1 value
Let HS = HMAC-SHA-1(K,C) // HS is a 20 byte string
Step 2: Generate a 4-byte string (Dynamic Truncation)
Let Sbits = DT(HS) // DT, defined in Section 6.3.1
// returns a 31 bit string
Step 3: Compute an HOTP value
Let Snum = StToNum(S) // Convert S to a number in
0...2^{31}-1
Return D = Snum mod 10^Digit // D is a number in the range
0...10^{Digit}-1
The Truncate function performs Step 2 and Step 3, i.e. the dynamic
truncation and then the reduction modulo 10^Digit. The purpose of
the dynamic offset truncation technique is to extract a 4-byte
dynamic binary code from a 160-bit (20-byte) HMAC-SHA1 result.
DT(String) // String = String[0]...String[19]
Let OffsetBits be the low order four bits of String[19]
Offset = StToNum(OffSetBits) // 0 <= OffSet <= 15
Let P = String[OffSet]...String[OffSet+3]
Return the Last 31 bits of P
The reason for masking the most significant bit of P is to avoid
confusion about signed vs. unsigned modulo computations. Different
processors perform these operations differently, and masking out
the signed bit removes all ambiguity.

Implementations MUST extract a 6-digit code at a minimum and
possibly 7 and 8-digit code. Depending on security requirements,
Digit = 7 or more SHOULD be considered in order to extract a longer
HOTP value.
The following paragraph is an example of using this technique for
Digit = 6, i.e. that a 6-digit HOTP value is calculated from the
HMAC value.
5.4 Example of HOTP computation for Digit = 6
The following code example describes the extraction of a dynamic
binary code given that hmac_result is a byte array with the
HMAC-SHA1 result:
int offset = hmac_result[19] & 0xf ;
int bin_code = (hmac_result[offset] & 0x7f) << 24
| (hmac_result[offset+1] & 0xff) << 16
| (hmac_result[offset+2] & 0xff) << 8
| (hmac_result[offset+3] & 0xff) ;
SHA-1 HMAC Bytes (Example)
-------------------------------------------------------------
| Byte Number |
-------------------------------------------------------------
|00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15|16|17|18|19|
-------------------------------------------------------------
| Byte Value |
-------------------------------------------------------------
|1f|86|98|69|0e|02|ca|16|61|85|50|ef|7f|19|da|8e|94|5b|55|5a|
-------------------------------***********----------------++|
* The last byte (byte 19) has the hex value 0x5a.
* The value of the lower four bits is 0xa (the offset value).
* The offset value is byte 10 (0xa).
* The value of the 4 bytes starting at byte 10 is 0x50ef7f19,
which is the dynamic binary code DBC1
* The MSB of DBC1 is 0x50 so DBC2 = DBC1 = 0x50ef7f19
* HOTP = DBC2 modulo 10^6 = 872921.
We treat the dynamic binary code as a 31-bit, unsigned, big-endian
integer; the first byte is masked with a 0x7f.
We then take this number modulo 1,000,000 (10^6) to generate the
6-digit HOTP value 872921 decimal.
6. Security Considerations
Any One-Time Password algorithm is only as secure as the
application and the authentication protocols that implement it.

Therefore, this section discusses the critical security
requirements that our choice of algorithm imposes on the
authentication protocol and validation software.
The parameters T and s discussed in this section have a significant
impact on the security - further details in Section 7 elaborate on
the relations between these parameters and their impact on the
system security.
It is also important to remark that the HOTP algorithm is not a
substitute for encryption and does not provide for the privacy of
data transmission. Other mechanisms should be used to defeat
6.1 Authentication Protocol Requirements
We introduce in this section some requirements for a protocol P
implementing HOTP as the authentication method between a prover and
a verifier.
RP1 - P MUST be two-factor, i.e. something you know (secret code
such as a Password, Pass phrase, PIN code, etc.) and something you
have (token). The secret code is known only to the user and usually
entered with the one-time password value for authentication purpose
(two-factor authentication).
RP2 - P SHOULD NOT be vulnerable to brute force attacks. This
implies that a throttling/lockout scheme is RECOMMENDED on the
validation server side.
RP3 - P SHOULD be implemented with respect to the state of the art
in terms of security, in order to avoid the usual attacks and risks
associated with the transmission of sensitive data over a public
network (privacy, replay attacks, etc.)
6.2 Validation of HOTP values
The HOTP client (hardware or software token) increments its counter
and then calculates the next HOTP value HOTP-client. If the value
received by the authentication server matches the value calculated
by the client, then the HOTP value is validated. In this case, the
server increments the counter value by one.
If the value received by the server does not match the value
calculated by the client, the server initiate the resynch protocol
(look-ahead window) before it requests another pass.
If the resynch fails, the server asks then for another
authentication pass of the protocol to take place, until the
maximum number of authorized attempts is reached.

If and when the maximum number of authorized attempts is reached,
the server SHOULD lock out the account and initiate a procedure to
inform the user.
6.3 Bi-directional Authentication
Interestingly enough, the HOTP client could also be used to
authenticate the validation server, claiming that it is a genuine
entity knowing the shared secret.
Since the HOTP client and the server are synchronized and share the
same secret (or a method to recompute it) a simple 3-pass protocol
could be put in place:
1- The end user enter the TokenID and a first OTP value OTP1;
2- The server checks OTP1 and if correct, sends back OTP2;
3- The end user checks OTP2 using his HOTP device and if correct,
uses the web site.
Obviously, as indicated previously, all the OTP communications have
to take place over secure https (SSL) connections.
6.4 Throttling at the server
Truncating the HMAC-SHA1 value to a shorter value makes a brute
force attack possible. Therefore, the authentication server needs
to detect and stop brute force attacks.
We RECOMMEND setting a throttling parameter T, which defines the
maximum number of possible attempts for One-Time-Password
validation. The validation server manages individual counters per
HOTP device in order to take note of any failed attempt. We
RECOMMEND T not to be too large, particularly if the
resynchronization method used on the server is window-based, and
the window size is large. T SHOULD be set as low as possible, while
still ensuring usability is not significantly impacted.
Another option would be to implement a delay scheme to avoid a
brute force attack. After each failed attempt A, the authentication
server would wait for an increased T*A number of seconds, e.g. say
T = 5, then after 1 attempt, the server waits for 5 seconds, at the
second failed attempt, it waits for 5*2 = 10 seconds, etc.
The delay or lockout schemes MUST be across login sessions to
prevent attacks based on multiple parallel guessing techniques.
6.5 Resynchronization of the counter
Although the server's counter value is only incremented after a
successful HOTP authentication, the counter on the token is
incremented every time a new HOTP is requested by the user. Because
of this, the counter values on the server and on the token might be
out of synchronization.

We RECOMMEND setting a look-ahead parameter s on the server, which
defines the size of the look-ahead window. In a nutshell, the
server can recalculate the next s HOTP-server values, and check
them against the received HOTP-client.
Synchronization of counters in this scenario simply requires the
server to calculate the next HOTP values and determine if there is
a match. Optionally, the system MAY require the user to send a
sequence of (say 2, 3) HOTP values for resynchronization purpose,
since forging a sequence of consecutive HOTP values is even more
difficult than guessing a single HOTP value.
The upper bound set by the parameter s ensures the server does not
go on checking HOTP values forever (causing a DoS attack) and also
restricts the space of possible solutions for an attacker trying to
manufacture HOTP values. s SHOULD be set as low as possible, while
still ensuring usability is not impacted.
6.6 Management of Shared Secrets
The operations dealing with the shared secrets used to generate and
verify OTP values must be performed securely, in order to mitigate
risks of any leakage of sensitive information. We describe in this
section different modes of operations and techniquest to perform
these different operations with respect of the state of the art in
terms of data security.
We can consider two different avenues for generating and storing
(securely) shared secrets in the Validation system:
* Deterministic Generation: secrets are derived from a master
seed, both at provisioning and verification stages and generated
on-the-fly whenever it is required;
* Random Generation: secrets are generated randomly at
provisioning stage, and must be stored immediately and kept secure
during their life cycle.
Deterministic Generation
------------------------
A possible strategy is to derive the shared secrets from a master
secret. The master secret will be stored at the server only. A
tamper resistant device MUST be used to store the master key and
derive the shared secrets from the master key and some public
information. The main benefit would be to avoid the exposure of the
shared secrets at any time and also avoid specific requirements on
storage, since the shared secrets could be generated on-demand when
needed at provisioning and validation time.
We distinguish two different cases:
- A single master key MK is used to derive the shared secrets;
each HOTP device has a different secret, K_i = SHA-1 (MK,i)
where i stands for a public piece of information that
identifies uniquely the HOTP device such as a serial number, a

token ID, etc.; obviously, this is in the context of an
application or service - different application or service
providers will have different secrets and settings;
- Several master keys MK_i are used and each HOTP device stores a
set of different derived secrets, {K_i,j = SHA-1(MK_i,j)} where
j stands for a public piece of information identifying the
device. The idea would be to store ONLY the active master key
at the validation server, in the HSM, and keep in a safe place,
using secret sharing methods such as [Shamir] for instance. In
this case, if a master secret MK_i is compromised, then it is
possible to switch to another secret without replacing all the
devices.
The drawback in the deterministic case is that the exposure of the
master secret would obviously enable an attacker to rebuild any
shared secret based on correct public information. The revocation
of all secrets would be required, or switching to a new set of
secrets in the case of multiple master keys.
On the other hand, the device used to store the master key(s) and
generate the shared secrets MUST be tamper resistant. Furthermore,
the HSM will not be exposed outside the security perimeter of the
validation system, therefore reducing the risk of leakage.
Random Generation
-----------------
The shared secrets are randomly generated. We RECOMMEND to follow
the recommendations in [RFC1750] and to select a good and secure
random source for generating these secrets. A (true) random
generator requires a naturally occurring source of randomness.
Practically, there are two possible avenues to consider for the
generation of the shared secrets:
* Hardware-based generators: they exploit the randomness which
occurs in physical phenomena. A nice implementation can be based on
oscillators, and built in such ways that active attacks are more
difficult to perform.
* Software-based generators: designing a good software random
generator is not an easy task. A simple, but efficient,
implementation should be based on various sources, and apply to the
sampled sequence a one-way function such as SHA-1.
We RECOMMEND to select proven products, being hardware or software
generators for the computation of shared secrets.
We also RECOMMEND storing the shared secrets securely, and more
specifically encrypting the shared secrets when stored using
tamper-resistant hardware encryption, and exposing them only when
required: e.g. the shared secret is decrypted when needed to verify
an HOTP value, and re-encrypted immediately to limit exposure in
the RAM for a short period of time. The data store holding the

shared secrets MUST be in a secure area, to avoid as much as
possible direct attack on the validation system and secrets
database.
Particularly, access to the shared secrets should be limited to
programs and processes required by the validation system only. We
will not elaborate on the different security mechanisms to put in
place, but obviously, the protection of shared secrets is of the
uttermost importance.
7. HOTP Algorithm Security: Overview
The conclusion of the security analysis detailed in the Appendix
section is that, for all practical purposes, the outputs of the
dynamic truncation (DT) on distinct counter inputs are uniformly
and independently distributed 31-bit strings.
The security analysis then details the impact of the conversion
from a string to an integer and the final reduction modulo
10^Digit, where Digit is the number of digits in an HOTP value.
The analysis demonstrates that these final steps introduce a
negligible bias, which does not impact the security of the HOTP
algorithm, in the sense that the best possible attack against the
HOTP function is the brute force attack.
Assuming an adversary is able to observe numerous protocol
exchanges and collect sequences of successful authentication
values. This adversary, trying to build a function F to generate
HOTP values based on his observations, will not have a significant
advantage over a random guess.
The logical conclusion is simply that is best strategy will once
again be to perform a brute force attack to enumerate and try all
the possible values.
Considering the security analysis in the Appendix section of this
document, without loss of generality, we can approximate closely
the security of the HOTP algorithm by the following formula:
Sec = sv/10^Digit
Where:
- Sec is the probability of success of the adversary
- s stands for the look-ahead synchronization window size;
- v stands for the number of verification attempts;
- Digit stands for the number of digits in HOTP values.
Obviously, we can play with s, T (the Throttling parameter that
would limit the number of attempts by an attacker) and Digit until
achieving a certain level of security, still preserving the system
usability.

8. Composite Shared Secrets
It may be desirable to include additional authentication factors in
the shared secret K. These additional factors can consist of any
data known at the token but not easily obtained by others. Examples
of such data include:
* PIN or Password obtained as user input at the token
* Phone number
* Any unique identifier programmatically available at the token
In this scenario the composite shared secret K is constructed
during the provisioning process from a random seed value combined
with one or more additional authentication factors. The server
could either build on-demand or store composite secrets - in any
case, depending on implementation choice, the token only stores the
seed value. When the token performs the HOTP calculation it
computes K from the seed value and the locally derived or input
values of the other authentication factors.
The use of composite shared secrets can strengthen HOTP based
authentication systems through the inclusion of additional
authentication factors at the token. To the extent that the token
is a trusted device this approach has the further benefit of not
requiring exposure of the authentication factors (such as the user
input PIN) to other devices.
9. IANA Considerations
This document has no actions for IANA.
10. Conclusion
This draft describes HOTP, a HMAC-based One-Time Password
algorithm. It also recommends the preferred implementation and
related modes of operations for deploying the algorithm.
The draft also exhibits elements of security and demonstrates that
the HOTP algorithm is practical and sound, the best possible attack
being a brute force attack that can be prevented by careful
implementation of countermeasures in the validation server.
Eventually, several enhancements have been proposed, in order to
improve security if needed for specific applications.
11. Acknowledgements
The authors would like to thank Siddharth Bajaj, Alex Deacon, Loren
Hart and Nico Popp for their help during the conception and
redaction of this document.
12. Contributors

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Appendix A - HOTP Algorithm Security: Detailed Analysis
The security analysis of the HOTP algorithm is summarized in this
section. We first detail the best attack strategies, and then
elaborate on the security under various assumptions, the impact of
the truncation and some recommendations regarding the number of
digits.
We focus this analysis on the case where Digit = 6, i.e. an HOTP
function that produces 6-digit values, which is the bare minimum
recommended in this draft.
A.1 Definitions and Notations
We denote by {0,1}^l the set of all strings of length l.
Let Z_{n} = {0,.., n - 1}.
Let IntDiv(a,b) denote the integer division algorithm that takes
input integers a, b where a >= b >= 1 and returns integers (q,r)

the quotient and remainder, respectively, of the division of a by
b. (Thus a = bq + r and 0 <= r < b.)
Let H: {0,1}^k x {0,1}^c --> {0,1}^n be the base function that
takes a k-bit key K and c-bit counter C and returns an n-bit output
H(K,C). (In the case of HOTP, H is HMAC-SHA-1; we use this formal
definition for generalizing our proof of security)
A.2 The idealized algorithm: HOTP-IDEAL
We now define an idealized counterpart of the HOTP algorithm. In
this algorithm, the role of H is played by a random function that
forms the key.
To be more precise, let Maps(c,n) denote the set of all functions
mapping from {0,1}^c to {0,1}^n. The idealized algorithm has key
space Maps(c,n), so that a "key" for such an algorithm is a
function h from {0,1}^c to {0,1}^n. We imagine this key (function)
to be drawn at random. It is not feasible to implement this
idealized algorithm, since the key, being a function from is way
too large to even store. So why consider it?
Our security analysis will show that as long as H satisfies a
certain well-accepted assumption, the security of the actual and
idealized algorithms is for all practical purposes the same. The
task that really faces us, then, is to assess the security of the
idealized algorithm.
In analyzing the idealized algorithm, we are concentrating on
assessing the quality of the design of the algorithm itself,
independently of HMAC-SHA-1. This is in fact the important issue.
A.3 Model of Security
The model exhibits the type of threats or attacks that are being
considered and enables to asses the security of HOTP and
HOTP-IDEAL. We denote ALG as either HOTP or HOTP-IDEAL for the
purpose of this security analysis.
The scenario we are considering is that a user and server share a
key K for ALG. Both maintain a counter C, initially zero, and the
user authenticates itself by sending ALG(K,C) to the server. The
latter accepts if this value is correct.
In order to protect against accidental increment of the user
counter, the server, upon receiving a value z, will accept as long
as z equals ALG(K,i) for some i in the range C,...,C + s-1, where s
is the resynchronization parameter and C is the server counter. If
it accepts with some value of i, it then increments its counter to
i+ 1. If it does not accept, it does not change its counter value.
The model we specify captures what an adversary can do and what it
needs to achieve in order to "win." First, the adversary is assumed

to be able to eavesdrop, meaning see the authenticator transmitted
by the user. Second, the adversary wins if it can get the server to
accept an authenticator relative to a counter value for which the
user has never transmitted an authenticator.
The formal adversary, which we denote by B, starts out knowing
which algorithm ALG is being used, knowing the system design and
knowing all system parameters. The one and only thing it is not
given a priori is the key K shared between the user and the server.
The model gives B full control of the scheduling of events. It has
access to an authenticator oracle representing the user. By calling
this oracle, the adversary can ask the user to authenticate itself
and get back the authenticator in return. It can call this oracle
as often as it wants and when it wants, using the authenticators it
accumulates to perhaps "learn" how to make authenticators itself.
At any time, it may also call a verification oracle, supplying the
latter with a candidate authenticator of its choice. It wins if the
server accepts this accumulator.
Consider the following game involving an adversary B that is
attempting to compromise the security of an authentication
algorithm ALG: K x {0,1}^c --> R.
Initializations - A key K is selected at random from K, a counter C
is initialized to 0, and the Boolean value win is set to false.
Game execution - Adversary B is provided with the two following
oracles:
Oracle AuthO()
--------------
A = ALG(K,C)
C = C + 1
Return O to B
Oracle VerO(A)
--------------
i = C
While (i <= C + s - 1 and Win == FALSE) do
If A == ALG(K,i) then Win = TRUE; C = i + 1
Else i = i + 1
Return Win to B
AuthO() is the authenticator oracle and VerO(A) is the verification
oracle.
Upon execution, B queries the two oracles at will. Let Adv(B) be
the probability that win gets set to true in the above game. This
is the probability that the adversary successfully impersonates the
user.

Let N = 2^31, d = 6 and m = 10^d. If x is chosen at random from
Z_{N} (meaning, is a random 31-bit string), then reducing it to a
6-digit number by taking x mod m does not yield a random 6-digit
number.
Rather, x mod m is distributed as shown in the following table:
Values Probability that each appears as output
----------------------------------------------------------------
0,1,...,483647 2148/2^31 roughly equals to 1.00024045/10^6
483648,...,999999 2147/2^31 roughly equals to 0.99977478/10^6
If X is uniformly distributed over Z_{2^31} (meaning is a random
31-bit string) then the above shows the probabilities for different
outputs of X mod 10^6. The first set of values appear with
probability slightly greater than 10^-6, the rest with probability
slightly less, meaning the distribution is slightly non-uniform.
However, as the Figure indicates, the bias is small and as we will
see later, negligible: the probabilities are very close to 10^-6.
A.4.2 Brute force attacks
If the authenticator consisted of d random digits, then a brute
force attack using v verification attempts would succeed with
probability sv/10^Digit.
However, an adversary can exploit the bias in the outputs of HOTP-
IDEAL, predicted by Lemma 1, to mount a slightly better attack.
Namely, it makes authentication attempts with authenticators which
are the most likely values, meaning the ones in the range 0,...,r -
1, where (q,r) = IntDiv(2^31,10^Digit).
The following specifies an adversary in our model of security that
mounts the attack. It estimates the success probability as a
function of the number of verification queries.
For simplicity, we assume the number of verification queries is at
most r. With N = 2^31 and m = 10^6 we have r = 483,648, and the
throttle value is certainly less than this, so this assumption is
not much of a restriction.
Proposition 1
-------------
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Assume
s <= m. The brute-force attack adversary B-bf attacks HOTP using v
<= r verification oracle queries. This adversary makes no
authenticator oracle queries, and succeeds with probability
Adv(B-bf) = 1 - (1 - v(q+1)/2^31)^s

which is roughly equals to
sv * (q+1)/2^31
With m = 10^6 we get q = 2,147. In that case, the brute force
attack using v verification attempts succeeds with probability
Adv(B-bf) roughly = sv * 2148/2^31 = sv * 1.00024045/10^6
As this equation shows, the resynchronization parameter s has a
significant impact in that the adversary's success probability is
proportional to s. This means that s cannot be made too large
without compromising security.
A.4.3 Brute force attacks are the best possible attacks
A central question is whether there are attacks any better than the
brute force one. In particular, the brute force attack did not
attempt to collect authenticators sent by the user and try to
cryptanalyze them in an attempt to learn how to better construct
authenticators. Would doing this help? Is there some way to "learn"
how to build authenticators that result in a higher success rate
than given by the brute-force attack?
The following says the answer to these questions is no. No matter
what strategy the adversary uses, and even if it sees, and tries to
exploit, the authenticators from authentication attempts of the
user, its success probability will not be above that of the brute
force attack - this is true as long as the number of
authentications it observes is not incredibly large. This is
valuable information regarding the security of the scheme.
Proposition 2
-------------
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Let B
be any adversary attacking HOTP-IDEAL using v verification oracle
queries and a <= 2^c - s authenticator oracle queries. Then
Adv(B) < = sv * (q+1)/ 2^31
Note: This result is conditional on the adversary not seeing more
than 2^c - s authentications performed by the user, which is hardly
restrictive as long as c is large enough.
With m = 10^6 we get q = 2,147. In that case, Proposition 2 says
that any adversary B attacking HOTP-IDEAL and making v verification
attempts succeeds with probability at most
Equation 1
----------
sv * 2148/2^31 roughly = sv * 1.00024045/10^6

Meaning, B's success rate is not more than that achieved by the
brute force attack.
A.5 Security Analysis of HOTP
We have analyzed in the previous sections, the security of the
idealized counterparts HOTP-IDEAL of the actual authentication
algorithm HOTP. We now show that, under appropriate and
well-believed assumption on H, the security of the actual
algorithms is essentially the same as that of its idealized
counterpart.
The assumption in question is that H is a secure pseudorandom
function, or PRF, meaning that its input-output values are
indistinguishable from those of a random function in practice.
Consider an adversary A that is given an oracle for a function f:
{0,1}^c --> {0, 1}^n and eventually outputs a bit. We denote Adv(A)
as the prf-advantage of A, which represents how well the adversary
does at distinguishing the case where its oracle is H(K,.) from the
case where its oracle is a random function of {0,1}^c to {0,1}^n.
One possible attack is based on exhaustive search for the key K. If
A runs for t steps and T denotes the time to perform one
computation of H, its prf-advantage from this attack turns out to
be (t/T)2^-k . Another possible attack is a birthday one [PrOo],
whereby A can attain advantage p^2/2^n in p oracle queries and
running time about pT.
Our assumption is that these are the best possible attacks. This
translates into the following.
Assumption 1
------------
Let T denotes the time to perform one computation of H. Then if A
is any adversary with running time at most t and making at most p
oracle queries,
Adv(A) <= (t/T)/2^k + p^2/2^n
In practice this assumption means that H is very secure as PRF. For
example, given that k = n = 160, an attacker with running time 2^60
and making 2^40 oracle queries has advantage at most (about) 2^-80.
Theorem 1
---------
Suppose m = 10^Digit < 2^31, and let (q,r) = IntDiv(2^31,m). Let B
be any adversary attacking HOTP using v verification oracle
queries, a <= 2^c - s authenticator oracle queries, and running
time t. Let T denote the time to perform one computation of H. If
Assumption 1 is true then

Adv(B) <= sv * (q + 1)/2^31 + (t/T)/2^k + ((sv + a)^2)/2^n
In practice, the (t/T)2^-k + ((sv + a)^2)2^-n term is much smaller
than the sv(q + 1)/2^n term, so that the above says that for all
practical purposes the success rate of an adversary attacking HOTP
is sv(q + 1)/2^n, just as for HOTP-IDEAL, meaning the HOTP
algorithm is in practice essentially as good as its idealized
counterpart.
In the case m = 10^6 of a 6-digit output this means that an
adversary making v authentication attempts will have a success rate
that is at most that of Equation 1.
For example, consider an adversary with running time at most 2^60
that sees at most 2^40 authentication attempts of the user. Both
these choices are very generous to the adversary, who will
typically not have these resources, but we are saying that even
such a powerful adversary will not have more success than indicated
by Equation 1.
We can safely assume sv <= 2^40 due to the throttling and bounds on
s. So:
(t/T)/2^k + ((sv + a)^2)/2^n <= 2^60/2^160 + (2^41)^2/2^160
roughly <= 2^-78
which is much smaller than the success probability of Equation 1
and negligible compared to it.
Appendix B - SHA-1 Attacks
This sections addresses the impact of the recent attacks on SHA-1
on the security of the HMAC-SHA-1 based HOTP. We begin with some
discussion of the situation of SHA-1 and then discuss the relevance
to HMAC-SHA-1 and HOTP. Cited references are at the bottom of the
document.
B.1 SHA-1 status
A collision for a hash function h means a pair x,y of different
inputs such that h(x)=h(y). Since SHA-1 outputs 160 bits, a
birthday attack finds a collision in 2^{80} trials. (A trial means
one computation of the function.) This was thought to be the best
possible until Wang, Yin and Yu announced on February 15, 2005 that
they had an attack finding collisions in 2^{69} trials.
Is SHA-1 broken? For most practical purposes we would say probably
not, since the resources needed to mount the attack are huge. Here
is one way to get a sense of it: we can estimate it is about the
same as the time we would need to factor a 760-bit RSA modulus, and
this is currently considered out of reach.

Burr of NIST is quoted [Crack] as saying ``Large national
intelligence agencies could do this in a reasonable amount of time
with a few million dollars in computer time.'' However, the
computation may be out of reach of all but such well-funded
agencies.
One should also ask what impact finding SHA-1 collisions actually
has on security of real applications such as signatures. To exploit
a collision x,y to forge signatures, you need to somehow obtain a
signature of x and then you can forge a signature of y. How
damaging this is depends on the content of y: the y created by the
attack may not be meaningful in the application context. Also, one
needs a chosen-message attack to get the signature of x. This seems
possible in some contexts, but not others. Overall, it is not clear
the impact on the security of signatures is significant.
Indeed, one can read that SHA-1 is ``broken,'' [Sha1], that
encryption and SSL are ``broken'' [Res], in the press. The media
have a tendency to magnify events: it would hardly be interesting
to announce in the news that a team of cryptanalysts did very
interesting theoretical work in attacking SHA-1.
Cryptographers are excited too. But mainly because this is an
important theoretical breakthrough. Attacks can only get beter with
time: it is therefore important to monitor any progress in hash
functions cryptanalysis and be prepared for any really practical
break with a sound migration plan for the future.
B.2 HMAC-SHA-1 status
The new attacks on SHA-1 have no impact on the security of HMAC-
SHA-1. The best attack on the latter remains one needing a sender
to authenticate 2^{80} messages before an adversary can create a
forgery. Why?
HMAC is not a hash function. It is a message authentication code
(MAC) that uses a hash function internally. A MAC depends on a
secret key, while hash functions don't. What one needs to worry
about with a MAC is forgery, not collisions. HMAC was designed so
that collisions in the hash function (here SHA-1) do not yield
forgeries for HMAC.
Recall that HMAC-SHA-1(K,x) = SHA-1(K_o,SHA-1(K_i,x)) where the
keys K_o,K_i are derived from K. Suppose the attacker finds a pair
x,y such that SHA-1(K_i,x)=SHA-1(K_i,y). (Call this a hidden-key
collision.) Then if it can obtain the MAC of x (itself a tall
order), it can forge the MAC of y. (These values are the same.) But
finding hidden-key collisions is harder than finding collisions,
because the attacker does not know the hidden key K_i. All it may
have is some outputs of HMAC-SHA-1 with key K. To date there are no
claims or evidence that the recent attacks on SHA-1 extend to find
hidden-key collisions.

Historically, the HMAC design has already proven itself in this
regard. MD5 is considered broken in that collisions in this hash
function can be found relatively easily. But there is still no
attack on HMAC-MD5 better than the trivial 2^{64} time birthday
one. (MD5 outputs 128 bits, not 160.) We are seeing this strength
of HMAC coming into play again in the SHA-1 context.
B.3 HOTP status
Since no new weakness has surfaced in HMAC-SHA-1, there is no
impact on HOTP. The best attacks on HOTP remain those described in
the document, namely to try to guess output values.
The security proof of HOTP requires that HMAC-SHA-1 behave like a
pseudorandom function. The quality of HMAC-SHA-1 as a pseudorandom
function is not impacted by the new attacks on SHA-1, and so
neither is this proven guarantee.
Appendix C - HOTP Algorithm: Reference Implementation
/*
* OneTimePasswordAlgorithm.java
* OATH Initiative,
* HOTP one-time password algorithm
*
*/
/* Copyright (C) 2004, OATH. All rights reserved.
*
* License to copy and use this software is granted provided that it
* is identified as the "OATH HOTP Algorithm" in all material
* mentioning or referencing this software or this function.
*
* License is also granted to make and use derivative works provided
* that such works are identified as
* "derived from OATH HOTP algorithm"
* in all material mentioning or referencing the derived work.
*
* OATH (Open AuTHentication) and its members make no
* representations concerning either the merchantability of this
* software or the suitability of this software for any particular
* purpose.
*
* It is provided "as is" without express or implied warranty
* of any kind and OATH AND ITS MEMBERS EXPRESSELY DISCLAIMS
* ANY WARRANTY OR LIABILITY OF ANY KIND relating to this software.
*
* These notices must be retained in any copies of any part of this
* documentation and/or software.
*/
package org.openauthentication.otp;

We introduce in this section several enhancements to the HOTP
algorithm. These are not recommended extensions or part of the
standard algorithm, but merely variations that could be used for
customized implementations.
E.1 Number of Digits
A simple enhancement in terms of security would be to extract more
digits from the HMAC-SHA1 value.
For instance, calculating the HOTP value modulo 10^8 to build an
8-digit HOTP value would reduce the probability of success of the
adversary from sv/10^6 to sv/10^8.
This could give the opportunity to improve usability, e.g. by
increasing T and/or s, while still achieving a better security
overall. For instance, s = 10 and 10v/10^8 = v/10^7 < v/10^6 which
is the theoretical optimum for 6-digit code when s = 1.
E.2 Alpha-numeric Values
Another option is to use A-Z and 0-9 values; or rather a subset of
32 symbols taken from the alphanumerical alphabet in order to avoid
any confusion between characters: 0, O and Q as well as l, 1 and I
are very similar, and can look the same on a small display.
The immediate consequence is that the security is now in the order
of sv/32^6 for a 6-digit HOTP value and sv/32^8 for an 8-digit HOTP
value.
32^6 > 10^9 so the security of a 6-alphanumeric HOTP code is
slightly better than a 9-digit HOTP value, which is the maximum
length of an HOTP code supported by the proposed algorithm.
32^8 > 10^12 so the security of an 8-alphanumeric HOTP code is
significantly better than a 9-digit HOTP value.
Depending on the application and token/interface used for
displaying and entering the HOTP value, the choice of alphanumeric
values could be a simple and efficient way to improve security at a
reduced cost and impact on users.
E.3 Sequence of HOTP values
As we suggested for the resynchronization to enter a short sequence
(say 2 or 3) of HOTP values, we could generalize the concept to the
protocol, and add a parameter L that would define the length of the
HOTP sequence to enter.
Per default, the value L SHOULD be set to 1, but if security needs
to be increased, users might be asked (possibly for a short period
of time, or a specific operation) to enter L HOTP values.

This is another way, without increasing the HOTP length or using
alphanumeric values to tighten security.
Note: The system MAY also be programmed to request synchronization
on a regular basis (e.g. every night, or twice a week, etc.) and to
achieve this purpose, ask for a sequence of L HOTP values.
E.4 A Counter-based Re-Synchronization Method
In this case, we assume that the client can access and send not
only the HOTP value but also other information, more specifically
the counter value.
A more efficient and secure method for resynchronization is
possible in this case. The client application will not send the
HOTP-client value only, but the HOTP-client and the related
C-client counter value, the HOTP value acting as a message
authentication code of the counter.
Resynchronization Counter-based Protocol (RCP)
----------------------------------------------
The server accepts if the following are all true, where C-server is
its own current counter value:
1) C-client >= C-server
2) C-client - C-server <= s
3) Check that HOTP-client is valid HOTP(K,C-Client)
4) If true, the server sets C to C-client + 1 and client is
authenticated
In this case, there is no need for managing a look-ahead window
anymore. The probability of success of the adversary is only v/10^6
or roughly v in one million. A side benefit is obviously to be able
to increase s "infinitely" and therefore improve the system
usability without impacting the security.
This resynchronization protocol SHOULD be use whenever the related
impact on the client and server applications is deemed acceptable.
E.5 Data Field
Another interesting option is the introduction of a Data field,
that would be used for generating the One-Time password values:
HOTP (K, C, [Data]) where Data is an optional field that can be the
concatenation of various pieces of identity-related information -
e.g. Data = Address | PIN.
We could also use a Timer, either as the only moving factor or in
combination with the Counter - in this case, e.g. Data = Timer,
where Timer could be the UNIX-time (GMT seconds since 1/1/1970)
divided by some factor (8, 16, 32, etc.) in order to give a
specific time step. The time window for the One-Time Password is

then equal to the time step multiplied by the resynchronization
parameter as defined before - e.g. if we take 64 seconds as the
time step and 7 for the resynchronization parameter, we obtain an
acceptance window of +/- 3 minutes.
Using a Data field opens for more flexibility in the algorithm
implementation, provided that the Data field is clearly specified.